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for QUIZ Oct. 27, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Suppose that f (x) is di erentiable everywhere and we know that f (2) = 1 and f (x) 2 for all x. a) What is the largest possible value for (4) f ? b) Show that f (x) has a root in [2, 4]. 2. Find the critical points of f (x) = 2x3 9x2 + 12x 2 and use the Second Derivative Test (if possible) to determine whether each corresponds to a local minimum or maximum.
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Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Oct. 27, 2008 1. Suppose that f (x) is dierentiable everywhere and we know that f (2) = 1 and f (x) 2 for all x. a) What is the largest possible value for f (4) ? b) Show that f (x) has a root in [2, 4]. Solution to 1a): By...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Oct. 30, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Sketch the curve y = 2x3 9x2 + 12x 2 . 2. Sketch the curve y= x2 1 1 . ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to theQUIZ for Oct. 30, 2008 1. Sketch the curve y = 2x3 9x2 + 12x 2 . Solution: y = 6x2 18x + 12, y = 12x 18. To get the critical points (numbers), we solve y = 0, getting 6x2 18x + 12 = 0, which is the same as 6(x 1)(x 2) = 0 yiel...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Nov. 3, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. A farmer has 3000 meters of fence, and needs to make a rectangular animal-shed, with a partition in the middle, parallel to one of the sides, using the same fencing materia...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Nov. 3, 2008 1. A farmer has 3000 meters of fence, and needs to make a rectangular animal-shed, with a partition in the middle, parallel to one of the sides, using the same fencing material. What are the dimensions of the r...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Nov. 6, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Evaluate x2 3x + 1 x+ e2x x lim . 2. Evaluate ln x x x lim . ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to theQUIZ for Nov. 6, 2008 1. Evaluate x2 3x + 1 x+ e2x x lim . Solution of 1: We use LHpital twice: o x2 3x + 1 2x 3 2 = lim = lim 2x x 2x 1 x+ x+ 2e x+ 4e2x e lim Now is the time to plug-in, and we get: 2 1 2 = = =0 2x x+ 4e 4e...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Nov. 10, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) d2 y dx2 dy dx (0) 1. Find y = y(x) if = 6x, = 0 and y(0) = 0. 2. Evaluate the following limit-Riemann sums by any method you like. n n lim i=1 3i n i n 2 1 n . ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Nov. 10, 2008 1. Find y = y(x) if d2 y dx2 = 6x, dy dx , dy dx (0) = 0 and y(0) = 0. Solution to 1: To get we take the anti-derivative of 6x: y (x) = 6x dx = 6x2 = 3x2 + C 2 . To nd C, we plug-in x = 0 and get, on the ...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Nov. 13, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Find 10 100 x2 dx 10 . 2. Using graphic analysis, nd the denite integral 1 1 x5 dx x4 + 3x2 + 1 (You must justify your answer). ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Nov. 13, 2008 1. Find 10 100 x2 dx 10 . Solution to 1: By the area denition of the denite integral, this is the area under the curve y = 100 x2 , above the interval [10, 10]. But this is a semi-circle of radius 100 ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Nov. 17, 2008 1. Use the Linear Approximation to approximate 49.1. Solution to 1: We are supposed to use the formula L(x) = f (a) + f (a)(x a) , First, we have to decide on f (x). This is f (x) = 49 = 7 is nice, it is ...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Nov. 24, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Compute ecos 2x sin 2x dx 2. Compute /4 sin4 2x cos 2x dx 0 ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Nov. 24, 2008 1. 1 4 1 1 3 x+ 3 + x x dx Sol. to 1: First, translate it into power notation: 4 3x1/2 + x3 + x1/2 dx 1 Now, integrate, piece-by-piece, using xn+1 +C . n+1 (except, for this kind of problems, you dont b...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Nov. 25, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Compute : 3 0 x2 dx +9 . 2. Compute 2 5x dx 0 . ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Nov. 25, 2008 1. Compute ecos 2x sin 2x dx . Sol. to 1: The natural substitution is u = cos 2x (note: not u = 2x). Dierentiating, we get (by the chain rule) du = 2 sin 2x , dx So here is the dictionary u = cos 2x Performing...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Dec. 1, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Find the area of the region enclosed between y = x2 2x and y = x 2. ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Dec. 1, 2008 1. Compute : 3 0 x2 dx +9 . Sol. to 1: We make the substitution x = 3u. Then dx = 3du and dont forger to translate the limits! When x = 0, u = 0, when x = 3, u = 1. We have: 3 0 1 0 dx = 2+9 x 1 0 3du = ...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Dec. 4, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Find dy/dx by implicit dierentiation if x3 y + xy + xy 4 = 4x2 4 . ...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Dec. 4, 2008 1. Find the area of the region enclosed between y = x2 2x and y = x 2. Sol. of 1: We rst nd the points of intersection by setting them equal to each other x2 2x = x 2 , means x2 3x + 2 = 0 Factoring, gives...
Rutgers >> CALC >> 1 (Fall, 2008)
QUIZ for Dec. 8, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Verify that the function f (x) = x4 + 1 satises the hypothesis of the Mean Value Theorem on the closed interval [0, 2]. Then nd all numbers c that satisfy the conclusion of...
Rutgers >> CALC >> 1 (Fall, 2008)
Solutions to the QUIZ for Dec. 8, 2008 1. Verify that the function f (x) = x4 + 1 satises the hypothesis of the Mean Value Theorem on the closed interval [0, 2]. Then nd all numbers c that satisfy the conclusion of the Mean Value Theorem. Sol. to 1...
Rutgers >> CALC >> 3 (Fall, 2008)
First Announcement INTEGERS CONFERENCE 2009 In Celebration of the 65th Birthdays of Mel Nathanson and Carl Pomerance October 14-17, 2009 University of West Georgia Carrollton, GA The Editors of Integers: Electronic Journal of Combinatorial Number ...
Rutgers >> MATH >> 251 (Fall, 2008)
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Rutgers >> MATH >> 336 (Fall, 2008)
T genetically engineered cell he biotechnology indusstrains to enhance the yield of a try is expanding rapidly MICHAEL A. HENSON target product [3]. While most due to advances in industrial processes are based on understanding complex microbial cell...
Rutgers >> MATH >> 250 (Fall, 2008)
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Rutgers >> MATH >> 250 (Fall, 2008)
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Rutgers >> MATH >> 250 (Fall, 2008)
MATLAB Tutorial You need a small number of basic commands to start using MATLAB. This short tutorial describes those fundamental commands. You need to create vectors and matrices, to change them, and to operate with them. Those are all short high-lev...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #2.1 [The Tangent and Velocity Problems] By Doron Zeilberger Problem Type 2.1.1 : The point P (a, f (a) lies on the curve y = f (x). (a) If Q is the point (x, f (x), use your calculator to nd the slope of the secant line P Q (c...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout # 2.2 [The Limit of a Function] By Doron Zeilberger Problem Type 2.2.1 : Given a graph of a function, nd the limits from the left, limit from the right, limit (if it exists), at various points, as well as some function values. ...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout # 2.3 [Calculating Limits Using the Limit Laws] By Doron Zeilberger Problem Type 2.3.1: Evaluate the limit if it exists: lim f (x) xa Example Problem 2.3.1 : Evaluate the limit if it exists: x2 + 5x + 4 x4 x2 + 3x 4 lim Ste...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #2.4 [The Precise Denition of a Limit] By Doron Zeilberger Problem Type 2.4.1 : Use the given graph of f (x) to nd a number such that |f (x) f (a)| < whenever |x a| < . (A graph is given with dashed lines at x = a and y = f...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #2.5 [Continuity] By Doron Zeilberger Problem Type 2.5.1 : Explain why the function is discontinuous at x = a: f (x) = Expression(x), if x = a; Number, if x = a. Example Problem 2.5.1: Explain why the function is discontinuous...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #2.6 [Limits at Innity: Horizontal Asymptotes] By Doron Zeilberger Problem Type 2.6.1: Find T OP (x) x BOT T OM (x) lim where both T OP and BOT T OM are polynomials in x. Example Problem 2.6.1: x3 + 5x x 2x3 x2 + 4 lim , , St...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #2.7 [Tangents, Velocities, and Other Rates of Change] By Doron Zeilberger Problem Type 2.7.1 : Find an equation for the tangent line to the curve at the given point. y = f (a), (a, f (a). Example Problem 2.7.1: Find an equatio...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #2.8 [Derivatives] By Doron Zeilberger Problem Type 2.8.1 : Find f (a) if f (variable) = Expression(variable). Example Problem 2.8.1: Find f (a) if f (s) = s+3 2s + 1 Steps 1. This is exactly like Problem 4.1 (section 2.7), bu...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.1 [Derivatives of Polynomials and Exponential Functions] By Doron Zeilberger Problem Type 3.1.1 : Dierentiate the function y = Expression(x) where the expression in x is a sum of terms involving powers (sometimes disguised a...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.2 [The Product and Quotient Rule] By Doron Zeilberger Problem Type 3.2.1 : Dierentiate f (x) = Expression1 (x) Expression2 (x), where both expressions are easy to dierentiate from known rules. Example Problem 3.2.1: Dierent...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.3 [Rates of Change in the Natural and Social Sciences] By Doron Zeilberger Problem Type 3.3.1 : A particle moves according to a law of motion s = f (t), t > 0, where t is measured in some unit of time and s in some unit of d...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.4 [Derivatives of Trigonometric Functions] By Doron Zeilberger Problem Type 3.4.1 : Dierentiate an expression involving products and/or quotients of expressions containing trig functions. Example Problem 3.4.1: Dierentiate y...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.5 [The Chain Rule] By Doron Zeilberger Problem Type 3.5.1 : Write the composite function in the form f (g(x), Identify the inner function u = g(x) and the outer function y = f (u). Then nd the derivative dy/dx. Example Probl...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.6 [Implicit Dierentiation] By Doron Zeilberger Problem Type 3.6.1 : Find dy/dx by implicit dierentiation, where you know that Expression1 (x, y) = Expression2 (x, y). Example Problem 3.6.1: Find dy/dx by implicit dierentiati...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.7 [Higher Derivatives] By Doron Zeilberger Problem Type 3.7.1 : Find the rst and second derivatives of the function f (x) = Expression(x). Example Problem 3.7.1: Find the rst and second derivatives of the function f (x) = ta...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.8 [Derivatives of Logarithmic Functions] By Doron Zeilberger Problem Type 3.8.1 : Dierentiate the function f (x) = Expression(x), where the expression involves ln x or loga x. Example Problem 3.8.1: Dierentiate the function ...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.10 [Related Rates] By Doron Zeilberger Problem Type 3.10.1 : If F (x, y) = c and dy/dt = a, nd dx/dt when y = b. Example Problem 3.10.1: If x3 + y 3 = 9 and dy/dt = 6 nd dx/dt when y = 2. Steps 1. Find the corresponding valu...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #3.11 [Linear Approximation] Problem Type 3.11.1: Use the Linear Approximation to estimate f = f (a + h) f (a) for the given function f (x), for the given a and h. Example Problem 3.11.1: Use the Linear Approximation to estima...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #4.1 [Maximum and Minimum Values] By Doron Zeilberger Problem Type 4.1.1 : Find the critical numbers of the function f (x) = Expression(x). Example Problem 4.1.1: Find the critical numbers of the function f (x) = x3 + 3x2 24x....
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #4.2 [The Mean Value Theorem] By Doron Zeilberger Problem Type 4.2.1 : Verify that the function f (x) satises the hypothesis of the Mean Value Theorem on the interval [a, b]. Then nd all numbers c that satisfy the conclusion of...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #4.3 [How Derivatives Aect the Shape of a Graph] By Doron Zeilberger Problem Type 4.3.1 : Given a function f (x) = P olynomial(x), (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and mi...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #4.4 [Indeterminate Forms and LHspitals Rule] o By Doron Zeilberger Problem Type 4.4.1 : Given certain limits of certain functions, f (x), g(x), . . . at a designated point x = a, determine whether the limits (at that very same...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #4.5 [Summary of Curve Sketching] By Doron Zeilberger Problem Type 4.5.1 : Sketch the curve y = P olynomial(x). Example Problem 4.5.1: Sketch the curve y = x4 + 4x3 Steps 1. Find the rst and second derivatives dy/dx and d2 y/d...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #4.7 [Optimization Problems] By Doron Zeilberger Problem Type 4.7.1 : A farmer wants to fence an area of A square units and then divide it into n + 1 parts by placing n parallel fences parallel to one of the sides of the rectan...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #4.9 [Newtons Method] By Doron Zeilberger Problem Type 4.9.1 : Use Newtons method with the specied initial approximation x1 to nd x3 , the third approximation to the root of the given equation. (Give your answer to four decimal...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #4.10 [Antiderivatives] By Doron Zeilberger Problem Type 4.10.1 : Find the most general antiderivative of the function f (x). Example Problem 4.10.1: Find the most general antiderivative of the function f (x) = 5ex + 8 sec2 x. ...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #5.2 [The Denite Integral] By Doron Zeilberger Problem Type 5.2.1 : Use the denition of the integral b n f (x)dx = lim a n f (xi )x i=1 (where x = (b a)/n and xi = a + ix). to evaluate the integral b f (x)dx a Example Pr...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #5.3 [The Fundamental Theorem of Calculus] By Doron Zeilberger Problem Type 5.3.1 : Use Part 1 of the Fundamental Theorem of Calculus to nd the derivative of V ariable1 f (V ariable1 ) = N umber Expression(V ariable2 )d V ari...
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #5.4 [Indenite Integrals and the Net Change Theorem] By Doron Zeilberger Problem Type 5.4.1 : Find the general indenite integral f (V ar)d V ar . Example Problem 5.4.1: Find (1 t)(2 + t2 )d t Steps 1. Use algebra and/or trig....
Rutgers >> CALC >> 1 (Fall, 2008)
Dr. Zs Math151 Handout #5.5 [The Substitution Rule] By Doron Zeilberger Problem Type 5.5.1 : Evaluate the indenite integral COM P LICAT ED(V ar) dV ar . Example Problem 5.5.1: Evaluate ex 1 + ex dx . Steps 1. Try to nd a good u. Usually whats i...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #6.1 [Areas Between Curves] By Doron Zeilberger Problem Type 6.1a : Find the area of the region enclosed between y = Expr1 (x) and y = Expr2 (x), where they meet at two points. Example Problem 6.1a: Find the area of the region ...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #6.2 [Volumes] By Doron Zeilbegrer Problem Type 6.2a: Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis y = f (x), x = a, x = b, y = 0 Example Problem 6.2a: Find the volu...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #6.3 [Volumes by Cylindrical Shells] By Doron Zeilberger Problem Type 6.3a: Use the method of cylindrical shells to nd the volume generated by rotating the region bounded by the given curves about the y-axis. y = f (x) , y=0 , ...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #6.4 [Work] By Doron Zeilberger Problem Type 6.4a: A spring has natural length of a meters. If an F0 -Newton force is required to keep it stretched to b meters, how much work is needed to stretch it from c meters to d meters? E...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #6.5 [Average Value of a Function] By Doron Zeilberger Problem Type 6.5a: Find the average value of a function f (x) in the interval [a, b] and nd c such that f (c) = fave . Example Problem 6.5a: Find the average value of f (x)...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #7.1 [Integration by Parts] By Doron Zeilberger Problem Type 7.1a: Evaluate the intgeral Something(x) SomethingElse(x) dx Example Problem 7.1a: Evaluate the intgeral x cos x dx Steps 1. Start with a blank table u= v= u= v= ...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #7.2 [Trigonometric Integrals] By Doron Zeilberger Problem Type 7.2a: Integrate an odd power of a sine or cosine. sin2n+1 x dx OR cos2n+1 x dx (n = 1, 2, . . .) Example Problem 7.2a: Evaluate the integral sin3 x dx Steps 1. R...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #7.3 [Trigonometric Substitution] By Doron Zeilberger Problem Type 7.3a: Integrate expressions involving a is some number. Example Problem 7.3a: Evaluate the integral 1 x2 dx Steps 1. If a2 x2 shows up use the substitution x...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #7.4 [Integration of Rational Functions by Partial Fractions] By Doron Zeilberger Problem Type 7.4a: Evaluate the integral C1 x + C2 dx (x a)(x b) , where a, b, C1 , C2 are numbers. Example Problem 7.4a: Evaluate the integ...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #7.7 [Approximate Integration] By Doron Zeilberger Problem Type 7.7a: Use (a) The Trapezoid Rule, (b) The Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specied value of n b f (x) dx a , n =...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #7.8 [Improper Integrals] By Doron Zeilberger Problem Type 7.8a: Determine whether the following integral is convergent or divergent. Evaluate it if it is convegent. f (x) dx , a where f (x) is easy to integrate. Example Pro...
Rutgers >> CALC >> 2 (Fall, 2008)
Dr. Zs Math152 Handout #8.1 [Arc Length] By Doron Zeilberger Problem Type 8.1a: Find the length of the curve y = f (x) , axb . Example Problem 8.1a: Find the length of the curve y = ln(cos x) , Steps 1. Find the derivative f (x) 0 x /3 . Example ...
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